Operators Definition and 1000 Threads

All these concepts belong to the toolbox of physicists. I read them quite often on our forum and their usage is sometimes a bit confusing. Physicists learn how to apply them, but occasionally I get the impression, that the concepts behind them are forgotten. So what are they? Especially when it...

Hi at all On my math methods book, i came across the following Fredholm integ eq with separable ker: 1) φ(x)-4∫sin^2xφ(t)dt = 2x-pi With integral ends(0,pi/2) I do not know how to proceed, for the solution...

I'm a beginner in QFT, starting out with D. J. Griffiths' book in this topic. I have a question on the operators used in QM. What are operators? What is the physical significance of operators? I can understand that ##\frac {d}{dt}## to be an operator, but how can there be a total energy...

If we consider a measurement of a two level quantum system made by using a probe system followed then by a von Neumann measurement on the probe, how could we determine the unitary operator that must be applied to this system (and probe) to accomplish the given measurement operators.

Hi All, Perhaps I am missing something. Schrodinger equation is HPsi=EPsi, where H is hamiltonian = sum of kinetic energy operator and potential energy operator. Kinetic energy operator does not commute with potential energy operator, then how come they share the same wave function Psi? The...

While at university, we went over logical operators for our electronic circuits lab. There was one that depended on the previous value which fascinated my deeply, but for some reason I can't remember it. I only vaguely remember it so I apologize if I mess up what actually happened. From what I...

Homework Statement Question: (With the following definitions here: - Consider ##L_0|x>=0## to show that ##m^2=\frac{1}{\alpha'}## - Consider ##L_1|x>=0 ## to conclude that ## 1+A-2B=0##- where ##d## is the dimension of the space ##d=\eta^{uv}\eta_{uv}## For the L1 operator I am able to get...

Hey guys, Am facing an issue, we know that x and y operators take the same form due to isotropy of space, but sir if we destroy the isotropy, then what form will it take? Can u pleases throw some light on this! Thanks in advance

Homework Statement Question: (With the following definitions here: ) - Consider ##L_0|x>=0## to show that ##m^2=\frac{1}{\alpha'}## - Consider ##L_1|x>=0 ## to conclude that ## 1+A-2B=0## - Consider ##L_2|x>=0 ## to conclude that ##d-4A-2B=0## - where ##d## is the dimension of the space...

I am reading a proof of why \left[ \hat{L}_x, \hat{L}_y \right ] = i \hbar \hat{L}_z Given a wavefunction \psi, \hat{L}_x, \hat{L}_y \psi = \left( -i\hbar \right)^2 \left( y \frac{\partial}{\partial z} - z \frac {\partial}{\partial y} \right ) \left (z \frac{\partial \psi}{\partial x} -...

Homework Statement Homework EquationsThe Attempt at a Solution 1.1st circle on the left : where did I miss a minus sign? 2. How to show that the last term is equal to 1? Thanks!

Hi. If I have an operator in matrix form eg. < i | x | j > then the matrix of the operator x2 is given by the square of the former matrix. This seems like common sense but how would i prove this using Dirac notation ? Thanks

Homework Statement STATEMENT ##\hat{H}=\int \frac{d^3k}{(2\pi)^2}w_k(\hat{a^+(k)}\hat{a(k)} + \hat{b^{+}(k)}\hat{b(k)})## where ##w_k=\sqrt{{k}.{k}+m^2}## The only non vanishing commutation relations of the creation and annihilation operators are: ## [\alpha(k),\alpha^{+}(p)] =(2\pi)^3...

Gauge symmetry is not a symmetry. It is a fake, a redundancy introduced by hand to help us keep track of massless particles in quantum field theory. All physical predictions must be gauge-independent...

Homework Statement Consider the free real scalar field \phi(x) satisfying the Klein-Gordon equation, write the Hamiltonian in terms of the creation/annihilation operators. Homework Equations Possibly the definition of the free real scalar field in terms of creation/annihilation operators...

Hello, I would just like some help clearing up some pretty basic things about hermitian operators and matricies. I am aware that operators can be represented by matricies. And I think I am right in saying that depending on the basis used the matrices will look different, but all our valid...

Hi guys, I have the following piece of code but I am not sure I understand if I get what it does correctly. static const unsigned int m_nBits = 6; static const unsigned int m_nRanges = 4; max = SOMENUMBER; if( max >= (1 << (m_nBits + m_nRanges - 1) ) ){ doStuff() } In fact I'm trying to...

Hi. I have come across the following statement - the eigenvalue equation for J+ is given by J+ | j m > = ħ √{(j+1)-m(m+1)} | j , m+1> My question is this - how can this be an eigenvalue equaton as the ket | j, m> has changed to | j , m+1> ? Surely the raising/lowering operators don't have...

Let D represent the differentiation of a single-parameter holomorphism, with respect to its parameter x. It's clear that for any sequence of holomorphisms g on x, sigma[k=0,inf](g[k](x)*D^k) is a linear operator on the space of holomorphisms. Is this a complete parameterization of the linear...

In classical mechanics, the Hamiltonian and the Lagrangian are Legendre transforms of each other. By analogy, in quantum mechanics and quantum field theory, the relationship between the Hamiltonian and the Lagrangian seems to be preserved. Where can I find a derivation of the Lagrangian...

I'm slowly working through Srednicki's QFT book and I had a question about section 3 (canonical quantization of scalar fields). At one point, he shows that the creation and annihilation operators ##a(\mathbf{k})## and ##a(\mathbf{k}')## are time-independent via the equation: $$a(\mathbf{k})...

In general I'm wondering if \lim_{x\to0} \left[\frac{d}{dy} \frac{d}{dx} f(x,y)\right] = \frac{d}{dy} \left[\lim_{x\to0} \frac{d}{dx} f(x,y)\right] holds true for all f(x,y). Thanks.

My question is, if I understand the question. For every "observable" physical corresponds a quantum operator. This operator can be represented as an infinite dimensional matrix in a Hilbert space. Only Hermitian matrices each may be quantum mechanical operators, and at the same time to an...

Hello everyone, I am wondering if the eigenstates of Hermitian operators, which represent possible wavefunctions representing the system, are always stationary wavefunctions, i.e. the deriving probability distribution function is always time invariant. I would think so since these eigenstates...

Hi Guys, at the moment I got a bit confused about the notation in some QM textbooks. Some say the operators should be symmetric, some say they should be self-adjoint (or in many cases hermitian what maybe means symmetric or maybe self-adjoint). Which condition do we need for our observables...

Homework Statement Homework Equations N/A The Attempt at a Solution I proved the first part of the question (first quote) and got stuck in the second (second quote). I defined Im(E1) as U and Im(E2) as W and proved that v=u+w where v ∈ V, u ∈ U and w ∈ W. After that however I got stuck at...

I need to start by saying that I'm not a physicist, nor a student of physics. I'm a translator, and my text is about rotations around the azimuthal nodal lines on the sphere. I need to find a name for a particular type of a rotation operator, which rotates the sphere around the x and y axes...

S. Weinberg says in his book, "The Quantum Theory of Fields Volume I", that Since electrons carry a charge, we would not like to mix annihilation and creation operators, so we might try to write the field as $$\psi(x)=\sum_{k}u_k (x)e^{-i\omega_k t}a_k$$ where ##u_k (x)e^{-i\omega_k t}## are a...

In Griffith and Sakurai QM book, spontaneous emission is treated as a closed system subject to time-dependent perturbation. Yet in quantum optics sponantanoues emission is treated as in the form master equation of density matrix. Even in two levels system where there is only one spontaneous...

The ladder operators of a simple harmonic oscillator which obey $$[H,a^{\dagger}]=\hbar\omega\ a^{\dagger}$$. --- I would like to see a proof of the relation $$\exp(-iHt)\exp(a^{\dagger})\exp(iHt)|0\rangle=\exp(a^{\dagger}e^{-i\omega t})|0\rangle\exp(i\omega t/2).$$ Thoughts?

Which of the operators T:C[0,1]\rightarrow C[0,1] are compact? $$(i)\qquad Tx(t)=\sum^\infty_{k=1}x\left(\frac{1}{k}\right)\frac{t^k}{k!}$$ and $$(ii)\qquad Tx(t)=\sum^\infty_{k=0}\frac{x(t^k)}{k!}$$ ideas for compactness of the operator: - the image of the closed unit ball is relatively...

When integrating terms including the imaginary unit i and operators like position and momentum, do you simply treat these as constants?

Let's say you have two operators A and B such that when they act on an eigenstate they yield a measurement of an observable quantity (so they're Hermitian). A and B do not commute, so they can't be measured simultaneously. My question is this: You have a matrix representation of A and B and...

The wavefunction describes the state of a system. When an operator 'acts on' the wavefunction are we saying, in layman's terms, that the operator is changing the state of the system?

I have been following a series of on-line lectures by Dr Physics A. He clearly describes what Hermitian operators for polarization and spin are and what they do. But when he gets to the position and momentum operators I am rather lost. They are no longer represented by square matrices. The...

In the dirac equation we have a term which is proportional to \alpha p . In the book they say that \alpha must be an hermitian operator in order for the Hamiltonian to be hermitian. As I understand, we require this because we want (\alpha p)^\dagger = \alpha p. But (\alpha p)^\dagger =...

Basically I've seen some expressions involving Hermitian Operators that I can't seem to justify, that others on the internet throw around like axiomatic starting points. (AB+BA)+ = (AB)++(BA)+? Why does this work? Assuming A&B are hermitian, I get why we can assume A+B is hermitian, but does...

I read that states are positive operators of unit trace - not elements of a vector space. Is it referring to quantum states or all classical states? I know operators are like minus, plus, square root and vectors are like rays in Hilbert space.. but why can't quantum states be vectors when in...

Extracted from 'At the frontiers of Physics, a handbook of QCD, volume 2', '...in the physical Bjorken ##x##-space formulation, an equivalent definition of PDFs can be given in terms of matrix elements of bi-local operators on the lightcone. The distribution of quark 'a' in a parent 'X'...

When I learned about operators, I learned <x> = ∫ Ψ* x Ψ dx, <p> = ∫ Ψ* (ħ/i ∂/∂x) Ψ dx. The book then told me the kinetic energy operator T = p2/2m = -ħ2/2m (∂2/∂x2) I am just think that why isn't it -ħ2/2m (∂/∂x)2 Put in other words, why isn't it the square of the derivative, but...

I am learning that operators corresponding to observable quantities are Hermitian since the eigenvalues are real. This makes sense (at least intuitively) and I have seen corresponding proofs of why eigenvalues of Hermitian operators are always real. That is fine. But are there any other types of...

I'm trying to check if Ehrenfest theorem is satisfied for this wave function, |Y>=(1/sqrt(2))*(|1>+|3>), where |1> and |3> are the ground and 1st exited state wave functions of a half harmonic oscillator. When I'm calculating the expectation values of x and p using annihilation creation...

Hello everyone, There's something I am not understanding in Hermitian operators. Could anyone explain why the momentum operator: px = -iħ∂/∂x is a Hermitian operator? Knowing that Hermitian operators is equal to their adjoints (A = A†), how come the complex conjugate of px (iħ∂/∂x) = px...

Homework Statement Particle is in a state with wave function \psi (r) = A z (x+y)e^{-\lambda r}. a) What is the probability that the result of the L_z measurement is 0? b) What are possilble results and what are their probabilities of a L^2 measurement? c) What are possilble results and what...

I'm reading McGreevy's lecture notes on holographic duality but I have two problems now: (See here!) 1) The author considers a matrix field theory for large N expansion. At first I thought its just a theory considered as a simple example and has nothing to do with the ## \mathcal N=4 ## SYM...

Homework Statement Vectors I1> and I2> create the orthonormal basis. Operator O is: O=a(I1><1I-I2><2I+iI1><2I-iI2><1I), where a is a real number. Find the matrix representation of the operator in the basis I1>,I2>. Find eigenvalues and eigenvectors of the operator. Check if the eigenvectors are...

Hi everybody. I have a (I gess rather silly) question. If I define [Jk,Ll]=iħΣmεklmLm, what would happen if I made [J, L]?. I gess it would be iħΣjεiijLi=0. Can someone please confirm this? Thanks for reading.

I am really struggling with one concept in my study of differential geometry where there seems to be a conflict among different textbooks. To set up the question, let M be a manifold and let (U, φ) be a chart. Now suppose we have a curve γ:(-ε,+ε) → M such that γ(t)=0 at a ∈ M. Suppose further...

While studying Ehrenfest's theorem I came across this formula for time-derivatives of expectation values. What I can't understand is why is position/momentum operator time-independent? What does it mean to be a time-dependent operator? Since position/momentum of a particle may change...

Hey all, i've found the following expression: How do they get that? They somehow used the kronecker delta Sum_k exp(i k (m-n))=delta_mn. But in the expression above, they're summing over i and not over r_i?? Best